In wireless communication, spatial correlation is the correlation between a signal's spatial direction and the average received signal gain. Theoretically, the performance of wireless communication systems can be improved by having multiple antennas at the transmitter and the receiver. The idea is that if the propagation channels between each pair of transmit and receive antennas are statistically independent and identically distributed, then multiple independent channels with identical characteristics can be created by precoding and be used for either transmitting multiple data streams or increasing the reliability (in terms of bit error rate). In practice, the channels between different antennas are often correlated and therefore the potential multi antenna gains may not always be obtainable.
Existence In an ideal communication scenario, there is a line-of-sight path between the transmitter and receiver that represents clear spatial channel characteristics. In urban cellular systems, this is seldom the case as base stations are located on rooftops while many users are located either indoors or at streets far from base stations. Thus, there is a non-line-of-sight multipath propagation channel between base stations and users, describing how the signal is reflected at different obstacles on its way from the transmitter to the receiver. However, the received signal may still have a strong spatial signature in the sense that stronger average signal gains are received from certain spatial directions. Spatial correlation means that there is a correlation between the received average signal gain and the angle of arrival of a signal. Rich multipath propagation decreases the spatial correlation by spreading the signal such that multipath components are received from many different spatial directions. Short antenna separations increase the spatial correlation as adjacent antennas will receive similar signal components. The existence of spatial correlation has been experimentally validated. Spatial correlation is often said to degrade the performance of multi antenna systems and put a limit on the number of antennas that can be effectively squeezed into a small device (as a mobile phone). This seems intuitive as spatial correlation decreases the number of independent channels that can be created by precoding, but is not true for all kinds of channel knowledge as described below.
Mathematical description In a narrowband flat-fading channel with N t {\displaystyle N_{t}} transmit antennas and N r {\displaystyle N_{r}} receive antennas (MIMO), the propagation channel is modeled as
y = H x + n {\displaystyle \mathbf {y} =\mathbf {H} \mathbf {x} +\mathbf {n} }
where y {\displaystyle \scriptstyle \mathbf {y} } and x {\displaystyle \scriptstyle \mathbf {x} } are the N r × 1 {\displaystyle \scriptstyle N_{r}\times 1} receive and N t × 1 {\displaystyle \scriptstyle N_{t}\times 1} transmit vectors, respectively. The N r × 1 {\displaystyle \scriptstyle N_{r}\times 1} noise vector is denoted n {\displaystyle \scriptstyle \mathbf {n} } . The i j {\displaystyle ij} th element of the N r × N t {\displaystyle \scriptstyle N_{r}\times N_{t}} channel matrix H {\displaystyle \scriptstyle \mathbf {H} } describes the channel from the j {\displaystyle j} th transmit antenna to the i {\displaystyle i} th receive antenna.
The common formula for the correlation matrix is:
R = E { v e c ( H ) ( v e c ( H ) ) H } {\displaystyle \mathbf {R} =E\left\{vec\left(\mathbf {H} \right)\left(vec\left(\mathbf {H} \right)\right)^{H}\right\}}
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